Questions: Differentiate. y = sqrt(x^2 - 8x + 25) dy/dx = □

Solution Steps

To differentiate the given function \( y = \sqrt{x^2 - 8x + 25} \), we can use the chain rule. The chain rule states that if you have a composite function \( y = f(g(x)) \), then the derivative \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \). Here, \( f(u) = \sqrt{u} \) and \( g(x) = x^2 - 8x + 25 \).

1. Identify the outer function \( f(u) = \sqrt{u} \) and the inner function \( g(x) = x^2 - 8x + 25 \).
2. Differentiate the outer function with respect to \( u \).
3. Differentiate the inner function with respect to \( x \).
4. Apply the chain rule to find \( \frac{dy}{dx} \).

Given the function: \[ y = \sqrt{x^2 - 8x + 25} \]

The inner function is: \[ g(x) = x^2 - 8x + 25 \] Differentiate \( g(x) \) with respect to \( x \): \[ g'(x) = 2x - 8 \]

The outer function is: \[ f(u) = \sqrt{u} \] Differentiate \( f(u) \) with respect to \( u \): \[ f'(u) = \frac{1}{2\sqrt{u}} \]

Using the chain rule: \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \] Substitute \( g(x) \) and \( g'(x) \): \[ \frac{dy}{dx} = \frac{1}{2\sqrt{x^2 - 8x + 25}} \cdot (2x - 8) \]

Simplify the derivative: \[ \frac{dy}{dx} = \frac{2x - 8}{2\sqrt{x^2 - 8x + 25}} \] \[ \frac{dy}{dx} = \frac{x - 4}{\sqrt{x^2 - 8x + 25}} \]

Final Answer